Neat exact sequences of abelian groups

ABSTRACT A subgroup of an abelian group is said to be neat if divisibility of it's element by every prime implies divisibility in this subgroup. Neatness is a generalization of pureness. First we establish fundamental properties of neat subgroups, then prove that the class of all short neat-exact sequences determined by neat subgroups is proper in Buchsbaum's sense. Neat projective groups are direct sums of cyclic groups of prime order and free group. Neat injective groups are direct sums of cyclic groups of prime order and divisible groups. Neat exact sequences are completely determined by projective (injective) property of cyclic groups of finite order. The subgroup of the group of extensions given by neat exact sequences is the frattini subgroup.